Find an equation of the tangent line to the curve y = 8x sin(x) at the point (π/2, 4π).

We'll need to find the derivative of the given function with respect to x and then evaluate the result at the value pi/2.

y = 8x sin(x) is a product:  y = [8x] * [sin x].  We must use the product rule for differentiation:

dy/dx = [8x][cos x] + [sin x][8]

Now substitute pi/2 in the above derivative.  We get:

(slope of tangent line) = dy/dx = [8pi/2][cos pi/2] + [sin pi/2][8]

The slope of the tangent line is      m = [8pi/2][0] + [1][8] = 8

Thus, the equation of the tangent line is y-4pi = 8[x-pi/2], or

2y-8pi = 8[2x-pi]